I think the problem here is much simpler than others have made it out to be.

Arriving happens instantaneously.

I arrived at 2 o'clock.

 

When will you arrive? (In an hour.)

Because of this, it's unnatural to ask how long something will take to arrive. Other verbs would suit better, like the other one in your example.

How long will this letter take to reach London?

 

How long will this letter take to get to London?

Your original sentence is understandable, and by no means would I stop you mid-sentence to correct you, but (because you asked), this use of arrive is a bit odd.

I would change it if I were editing a formal English document, for example.

As mentioned in other answers, at is also not the correct preposition to use. Usually, in is used with large bodies, like cities and countries. For example, I would arrive in Oslo, but at my hotel, since a hotel isn't miles and miles around.

I think the problem here is much simpler than others have made it out to be.

Arriving happens instantaneously.

I arrived at 2 o'clock.

When will you arrive? (In an hour.)

Because of this, it's unnatural to ask how long something will take to arrive. Other verbs would suit better, like the other one in your example.

How long will this letter take to reach London?

How long will this letter take to get to London?

Your original sentence is understandable, and by no means would I stop you mid-sentence to correct you, but (because you asked), this use of arrive is a bit odd.

I would change it if I were editing a formal English document, for example.

As mentioned in other answers, at is also not the correct preposition to use. Usually, in is used with large bodies, like cities and countries. For example, I would arrive in Oslo, but at my hotel, since a hotel isn't miles and miles around.